Preheat the oven to 425,Prepare the pie dough and line a 10-inch deep dish pie pan with half the dough.,Peel and core the apples and slice into eighths.,In a bowl, toss the apples with the lemon juice, sugar and cinnamon.,Stir in the tapioca.,Turn the filling into the pie pan and dot with butter.,Roll out the remaining dough into a circle.,______,Seal and crimp.,Brush the crust evenly with the egg wash.,Cut vents in the top crust by inserting a paring knife 1/4 inch through the dough.,Bake in the center of the oven for 20 minutes.,Reduce the heat to 350 and continue baking for 45 minutes more.,Cool for 1 hour before serving.,Serve with cheddar cheese.